It is known that the assumption that ``GCH first fails at \aleph_{\omega}''leads to large cardinals in ZFC. Gitik and Koepke have demonstrated that thisis not so in ZF: namely there is a generic cardinal-preserving extension of L(or any universe of ZFC + GCH in which all ZF axioms hold, the axiom of choicefails, GCH holds for all cardinals \aleph_n, but there is a surjection fromPowerSet(\aleph_{\omega}) onto {\lambda}, where {\lambda} is any previouslychosen cardinal in L greater than \aleph_{\omega}, for instance, \aleph_{\omega+17}. In other words, in such an extension GCH holds in proper sense for allcardinals \aleph_n but fails at \aleph_{\omega} in Hartogs' sense. The goal ofthis note is to analyse the system of automorphisms involved in the Gitik --Koepke proof.
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